Moving average Multifractional Processes with Random Exponent: lower bounds for local oscillations

Antoine Ayache (1) , Florent Bouly (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.

Mots clés: Fractional Brownian Motion; varying Hurst parameter; pointwise Hölder regularity; Itô integral

Domaines: Mathématiques [math]

Antoine Ayache, Florent Bouly. Moving average Multifractional Processes with Random Exponent: lower bounds for local oscillations. Stochastic Processes and their Applications, 2022. ⟨hal-03311329⟩